An alternative to Johns clipper plugin (thanks btw I didn't know about that one) - is the "Difference" tool in the processing toolbox, in combination with the "Edit features in place" mode.
Two Polygons - make the layer you want to remove the shape from the "Active" layer by selecting it like I have in my screen shot

Go to you processing toolbox, select the "Edit Feature in place" mode, and then search for the difference tool and select it.
In the difference tool select the layer which has the shape to be removed
.
The result will be your target layer is in an edit session with the object removed. Once you save the edit on that layer it will be permanent.
End result after clearing selection.